Hong Life CalculatorsUnderstanding the linear independence calculator — concept and principle
This calculator tests whether a set of vectors is linearly independent. If it is not, it finds a dependence relation, names the vectors that depend on earlier ones and gives the rank. Use it for linear algebra homework on bases, span and the invertibility of matrices.
How it is calculated
Put the vectors as columns of a matrix and row-reduce it.
- Independent exactly when every column has a pivot, i.e. rank = number of vectors.
- A column without a pivot is a combination of earlier pivot columns; the reduced form gives the coefficients of a dependence relation c₁v₁ + … + c_kv_k = 0.
- More vectors than their dimension (k > n) are always dependent.
The principle
Vectors are independent when none of them can be built from the others: the only way to combine them into the zero vector is to use all-zero coefficients. If some non-trivial combination gives zero, one vector can be solved for in terms of the rest, so it adds nothing new to the span. Row reduction exposes exactly this, because row operations keep the linear relations between columns.
Worked example
Default input: v₁ = (1, 2, 3), v₂ = (4, 5, 6), v₃ = (7, 8, 9).
- Row reduction gives pivots in columns 1 and 2 only: rank 2 with 3 vectors of dimension 3.
- Linearly independent? No — linearly dependent.
- Vector that depends on earlier ones: v3.
- Dependence relation: v₁ − 2·v₂ + v₃ = 0; check: (1, 2, 3) − (8, 10, 12) + (7, 8, 9) = (0, 0, 0).
How to read the result
Here v₃ = 2v₂ − v₁: the three vectors lie in one plane through the origin, so they span only a 2-dimensional space. A square matrix whose columns are dependent has determinant 0 and no inverse. Removing the dependent vector leaves a basis of the span.
Common mistakes
- Thinking vectors are independent because none is a multiple of another; with three or more vectors, dependence can involve several at once.
- Forgetting that any set containing the zero vector is dependent.
- Mixing up rows and columns when setting up the matrix.
- Trusting near-zero pivots from rounded decimals.
Frequently asked questions
How do I check if vectors are linearly independent?
Row-reduce the matrix with the vectors as columns; they are independent if every column has a pivot.
Are (1, 2, 3), (4, 5, 6), (7, 8, 9) independent?
No: v₁ − 2v₂ + v₃ = 0.
Can four vectors in ℝ³ be independent?
No, at most three vectors in ℝ³ can be independent.
Sources: A. Davis, P. Zachlin et al., Coordinated Linear Algebra (University of Nebraska–Lincoln), §2.4 Homogeneous Linear Systems · D. Margalit & J. Rabinoff, Interactive Linear Algebra, §2.7